Analysis of TMT Primary Mirror Control-Structure Interaction (SPIE 7017-41) Douglas MacMynowski (Caltech) Peter Thompson (Systems Tech.) Mark Sirota (TMT Observatory)
Control Problems TMT.SEN.PRE.08.046.REL01 2
Primary Mirror Control System (M1CS) Sensors measure relative displacement between segments, estimate segment position: Actuators: approach TBD, but candidates include Soft (e.g. voice-coil), stiffened by servo loop Hard (e.g. screw-type actuator), here I assume open loop Control Servo control loop (if soft actuators used), ~20 Hz bandwidth to obtain low frequency desired stiffness (see 7012-58) Global control loop: need ~1 Hz bandwidth to reject wind (see 7017-31) Mirror cell introduces structural coupling that affects both of these TMT.SEN.PRE.08.046.REL01 3
Outline Primary Mirror Control System Control-Structure Interaction TMT structural dynamics Actuator servo control loop Global M1 control loop TMT.SEN.PRE.08.046.REL01 4
Control-Structure Interaction (CSI) Design without knowledge of structure clear potential for problems Control (both global & servo) can couple into structural modes Potential for interacting control loops, e.g. each loop stable on its own, but the combination is unstable Aubrun & Lorell study on ASCIE in early 90 s: destabilization proportional to number of control loops Aubrun & Lorell Keck CSI analysis: 0.5 Hz maximum bandwidth Start with a really simple problem to build intuition (see paper), then apply to TMT M1CS It is the ratio of total segment mass n m to mirror cell mass M that determines stability (not n alone) TMT.SEN.PRE.08.046.REL01 5
M1 Dynamics f i m m m m x i k k k u i k z i Mirror cell 492 copies! TMT.SEN.PRE.08.046.REL01 6
Computational Simplification Use structural modes: good intuition for isolated mirror cell, but actual modes are not orthogonal when evaluated at segment loc ns. Use to show that coupling depends on total mass ratio μ=(n m)/m Analyze dynamics in Zernike basis Coupling converges with relatively few basis vectors More convenient to describe patterns of segment motion rather than individual segment motion Dynamics will be coupled; MIMO analysis is required Z 2,0 Z 2,+2 Z 2,-2 TMT.SEN.PRE.08.046.REL01 7
Telescope Structure Finite element model of full telescope without segments - Compliance is highest at low spatial frequency - Assume 0.5% damping 10-6 Structure (LR) Segment support Compliance (m/n) 10-7 10-8 12 3 4 5 6 7 8 9 10 TMT.SEN.PRE.08.046.REL01Zernike basis function radial degree8
Modeling Approach Telescope FEM has no segment dynamics (intentionally) Allows relevant segment dynamics to be added as needed for analysis using separate (detailed) FEM of segment assembly Segment model reduced to 33 modes for CSI analysis Telescope model and segment model connected at actuator nodes. Projected into Zernike basis Connect with stiff spring segment (w/ rigid body modes) cell TMT.SEN.PRE.08.046.REL01 9
Idealized Actuator Assumptions Soft (voice-coil) actuator: Local position (servo) loop between force and collocated displacement Offload spring in parallel f, y m k=1.6e5 N/m Potential to add damping in parallel Leads to a simple servo control m Hard actuator (e.g. screw + piezo): Perfect displacement device (output = command at all frequencies) Pure stiffness (1e7 N/m) with zero damping k=1e7 N/m d TMT.SEN.PRE.08.046.REL01 10
Servo open-loop TF on structure: Zernike basis Transfer function from voice-coil force to position In Zernike basis, mounted on structure Note maximum structure 10-4 resonance included is Z 0,0 36 Hz (static correction Z retained to 2000 modes, 1,+1 ~70 Hz) 10-6 Note that response on different basis functions is NOT orthogonal; multivariable stability robustness analysis is required! Response (m/n) 10-8 Z 1,-1 Z 2,+2 Z 2,-2 Z 2,0 Rigid Coupling to telescope structure Segment piston/tip/tilt, on offload spring Segment support modes 10 0 10 1 10 2 Frequency (Hz) TMT.SEN.PRE.08.046.REL01 11
Servo open-loop TF on structure: Zernike basis Transfer function from voice-coil force to position In Zernike basis, mounted on structure Note maximum structure 10-4 resonance included is Z 0,0 36 Hz (static correction Z retained to 2000 modes, 1,+1 ~70 Hz) 10-6 Note that response on different basis functions is NOT orthogonal; multivariable stability robustness analysis is required! Response (m/n) 10-8 Z 1,-1 Z 2,+2 Z 2,-2 Z 2,0 Rigid Coupling to telescope structure Segment piston/tip/tilt, on offload spring Segment support modes 10 0 10 1 10 2 Frequency (Hz) TMT.SEN.PRE.08.046.REL01 12
Servo Loop Design Two strategies: 1. No change to plant: difficult to simultaneously navigate CSI with telescope (most low frequency) and interaction with segment dynamics (high frequency) 10 8 2. Add passive damping in parallel to actuator: stabilizes 90 Hz+ segment dynamic modes Gain (N/m) 10 7 See P. Thompson et al. for details (SPIE 7012-xx) 10 6 10-1 10 0 10 1 10 2 Frequency (Hz) TMT.SEN.PRE.08.046.REL01 13
Multivariable (MIMO) Robustness Evaluation SISO analysis is not sufficient (doesn t capture coupling) Generalization is H : Want S < 2 (analogous to SISO GM/PM constraints) Plot the maximum singular value of multivariable sensitivity at each frequency Distance to critical point is 1+L = S(jω) -1 min 1+L >1/2 GM > 6dB, PM > 30 TMT.SEN.PRE.08.046.REL01 14
Servo Loop Robustness Robustness is limited by structural modes that project almost entirely onto Zernike radial degree 2 and 3 Accurately predicted using only 10 (not 492) basis functions σ max (S) 3.5 3 2.5 2 1.5 1 Current design not sufficiently robust! p=0 p 1 p 2 p 3 p 4 p 5 p 6 Max Sensitivity 3.5 3 2.5 2 1.5 1 S 10.5 Hz mode 0.5 0.5 0 0 0 10 20 30 40 50 0 1 2 3 4 5 6 Frequency (Hz) TMT.SEN.PRE.08.046.REL01 Max Zernike radial degree included 15
Global Loop Transfer function from commanded actuator displacement to mirror motion: Telescope structure modes are visible Global control loop doesn t depend (much) on hard vs soft actuator Response (m/m) 10 1 10 0 Z 2,+2 Z 2,-2 Z 2,0 Rigid 10.3 10.5 Soft actuator 13 Response (m/m) 10 2 10 1 10 0 Soft actuator Hard actuator 10 0 10 1 10 2 10 0 10 1 10 2 Frequency (Hz) Frequency (Hz) TMT.SEN.PRE.08.046.REL01 16 10-1
Global Control Design SISO analysis: With 2-pole roll-off at 7 Hz, maximum bandwidth ~ 2Hz (6 db gain margin) Even with hard actuator, limited by telescope structure, not 35 Hz segment resonance Loop transfer fn 10 1 10 0 10-1 10-2 Z 2,0 Z 2,-2 Z 2,2 Rigid Base Mag. 0.3, Freq. 14.8 10-3 10-1 10 0 10 1 10 2 Frequency (Hz) TMT.SEN.PRE.08.046.REL01 17
Global Control Loop MIMO Robustness Evaluation 1.5 Hz control bandwidth is achievable (Keck uses <0.1 Hz) Higher bandwidth achievable at higher spatial frequency: >2 Hz for radial degree > 4 2 Depends on assumed structural damping 1.5 Stability boundary predicted using 3 (not 492) basis functions No significant difference in global control between hard or soft actuators. σ max (S) 1 1.5 Hz control bandwidth: S =1.87 0.5 2 p 5 p=2 only Rigid base 0 0 5 10 15 Frequency (Hz) TMT.SEN.PRE.08.046.REL01 18
Conclusions Control-structure interaction limits the achievable bandwidth of primary mirror control system Zernike-basis analysis simplifies computation Local control loop (soft actuator): Challenging design, plausible with passive damping in parallel Global control loop: 1 Hz desired control bandwidth appears achievable Achievable bandwidth is the same for hard or soft actuators Depends on damping assumptions Errors in A matrix may also limit bandwidth TMT.SEN.PRE.08.046.REL01 19
Acknowledgments The authors gratefully acknowledge the support of the TMT partner institutions. They are the Association of Canadian Universities for Research in Astronomy (ACURA), the California Institute of Technology and the University of California. This work was supported as well by the Gordon and Betty Moore Foundation, the Canada Foundation for Innovation, the Ontario Ministry of Research and Innovation, the National Research Council of Canada, the Natural Sciences and Engineering Research Council of Canada, the British Columbia Knowledge Development Fund, the Association of Universities for Research in Astronomy (AURA) and the U.S. National Science Foundation. TMT.SEN.PRE.08.046.REL01 20